Waiting for affordable housing in New York City

1 Introduction

As the U.S. has shifted toward a knowledge-based economy, most large cities in the U.S. have experienced a surge in popularity. As a consequence, real estate prices and rental rates for housing have soared in many U.S. cities for the past two decades. These increases in the prices of housing can be explained by an inelastic supply of housing due to restrictive zoning laws combined with increasing demand of high-income households who prefer to live in attractive cities.1 These economic trends have created a shortage of affordable housing in the most expensive housing markets such as Boston, Los Angeles, New York, San Francisco, and Seattle. Many progressive politicians have included the expansion of affordable housing policies in their platforms.

The political popularity of affordable policies in these cities is in stark contrast to long term trends in the supply of affordable housing. To increase the supply of housing, local governments often need to change zoning laws. Landowners and housing developers reap substantial windfall gains from rezoning. Politicians can redistribute part of these windfall gains from rezoning to renters by mandating that a certain amount of new housing is offered at affordable rates. The key insight here is that most affordable housing policies are not necessarily motivated by efficiency gains, but are driven by political conveniences. It is well known that affordable housing policies create a mismatch and often reduce the incentives to increase housing supply. The political popularity of these policies simply reflect the wishes and needs of the majority of voters that tend to be low- and moderate-income renters.2

Despite the importance, prevalence, and political popularity of affordable housing policies in high-cost cities, there are few compelling dynamic models that allow us to quantify the benefits that arise from these policies to low- and moderate-income households.3 The objective of this paper is to fill this gap. We develop and estimate a new dynamic equilibrium model with heterogeneous households that is consistent with the observed market frictions, the presence of mobility or adjustment costs, the existence of long queues for public housing, and the need to search for a long time to obtain access to rent-stabilized housing. Our model, therefore, captures the existence of three different types of rental markets—public, regulated, and unregulated rental markets—and formalizes the dynamic incentives faced by households.

We assume that households can rent any amount of housing in the unregulated market given the prevailing market price per unit of housing services. However, moving is costly. This implies that a household faces some adjustment costs when moving to a new rental unit that better fits its needs. All households also have access to regulated or rent-stabilized housing. We assume that the rental price for rent-stabilized housing is significantly lower than the equivalent market price in the unregulated market. Since the demand for rent-stabilized units typically exceeds the supply, there are significant frictions in the rent-stabilized market. Finding a rent-stabilized apartment involves significant search efforts and luck. We capture these market frictions by endogenizing the probability that a household who is actively searching for rent-stabilized housing receives an offer to move into a stabilized unit. The search frictions and moving costs then create misallocations of housing units in rental markets.4

Low- and moderate-income households are also eligible for public housing assistance in our model if their incomes are below an exogenously determined threshold. The rent charged for public housing is a fixed percentage of household income. Hence, there is no price mechanism to ensure that public housing markets clear. Demand for public housing vastly exceeds the available supply in most affluent U.S. cities. Rationing is achieved by placing households on waitlists that allocate free units to households with the highest priority, that is, households that have waited for public housing the longest.5 Moreover, the housing authority does not evict households after they have lost their eligibility for housing aid. Consequently, these policies imply that public housing is not necessarily allocated to the most deserving low-income households.

Our model significantly improves over previous studies that have been based on static models. Any model that tries to capture search or queuing behavior needs to be dynamic. The main challenges relate to modeling the degree of forward-looking behavior and the beliefs that households hold with respect to the probabilities of obtaining access to public or affordable housing. Public and affordable housing also offers partial insurance against negative income shocks. Upon receiving a positive income shock, a household may consider moving out from public housing or rent-regulated housing. However, households need to take into consideration that they may be hit by negative income shock in the future and that it will take a long time to get back into public or affordable housing. These insurance aspects are potentially important components of the welfare gains associated with public or affordable housing.

We define and characterize a stationary equilibrium with search and rationing. The length of the waitlist for public housing and the probability of finding a rent-stabilized unit are all endogenously determined in the equilibrium of our model. Low- and moderate-income households prefer to live in public housing due to the large rent subsidy, which implies a large increase in numeraire consumption, and the relatively high quality of these housing units. Rent-stabilized housing appeals to a large range of low-, moderate-, and even high-income households due to the significant rental price discount relative to the unregulated market. Due to the existence of rationing in public housing and search frictions in rent-stabilized housing a fraction of low- and moderate-income households must also rent in the unregulated market in equilibrium. Our model can, therefore, explain the existence of long wait and search times and is consistent with the observed misallocations in public and rent-stabilized housing markets.

The parameters of our model can be identified based on the observed moments in the data. Our proof of identification is constructive and can be used to define a method of moments estimator. This estimator matches the sorting of households by income and family type among housing options and the average time spent in different housing markets. The estimator also matches the average rental payments for each housing type.6

Our empirical analysis focuses on New York City (NYC). While many cities in the U.S. and abroad face the challenge to provide an adequate supply of affordable housing, NYC has been at the center of the debate over the desirability of affordable housing policies. Studying the housing markets for low- and moderate-income households in NYC is promising for a variety of compelling reasons. First, NYC has the largest stock of rental apartments of all cities in the U.S. and is generally perceived to be one of the most expensive rental markets in the world. Second, New York City also has the largest stock of public housing units of all cities in the U.S. Finally, NYC is the only large city in the United States that has ever declared a housing emergency and has adopted strict rent-stabilization policies over an extended period of time.7 NYC, therefore, serves as a laboratory to explore the effectiveness and impact of a affordable housing policies.

Our empirical analysis is based on the 2011 sample of the New York City Housing and Vacancy Survey (NYCHVS). This survey provides comprehensive data about household and housing characteristics. In particular, we observe household income and family status, the time that the household has spent in the housing unit, as well as a large number of structural characteristics of the housing unit that the household occupies. In addition, it allows us to classify households as living in public housing, rent-stabilized housing, or unregulated housing. We implement our estimator focusing on Manhattan since waitlists for public housing in NYC are operated at the borough level. The data show that approximately of our sample of low- and moderate-income households lived in public housing communities in 2011. Fifty-seven percent of households lived in rent-stabilized units. The remaining rented in the unregulated housing market. At the time of the survey, households spent, on average, 16 years in public housing, 9.5 years in regulated housing, and only 4 years in unregulated housing. Not surprisingly, households in public housing are much poorer than households in rent-stabilized and unregulated housing.

Our model fits the sorting of households by income among the three housing options. It captures differences in rental prices as well as time spent in the housing units. Our model allows us to quantify the importance of search and queuing frictions, as well as adjustment costs. Our findings suggest that all three types of frictions are highly relevant to understand rental markets in NYC. We find that rental prices for stabilized housing are approximately lower than prices in the unregulated market in Manhattan. This significant discount explains the popularity of rent-stabilized units. The probability of finding a rent-stabilized unit is approximately per year for a relatively affordable, low-quality unit and for a more expensive, high-quality unit. The average wait time for public housing is approximately 19 years.

Policymakers and politicians have struggled to find a response to the increasing demands of low- and moderate-income voters to increase the supply of affordable housing. Our model provides a compelling explanation of why these policies have been so popular at the ballot box with the vast majority of urban renters. Our analysis shows that low-quality units of affordable housing are attractive for most low- and moderate-income households. These households gain up to from having access to stabilized units. For high-quality units, the results are even more striking. High-quality units are attractive for moderate- and high-income households and generate welfare gains of up to .8 Given these large benefits associated with affordable housing, it is not surprising that rent-stabilization and affordable housing policies are popular, not only with low- and moderate-income households, but with the vast majority of all urban renters in NYC.

We then evaluate the impact of increasing the supply of affordable housing on the distribution of renters' welfare. We find that a increase in the supply of affordable improves welfare for all renters since the wait and search times decrease. The average welfare gains associated with a permanent increase in the stock of affordable housing are approximately . Of course, these findings do not imply that these policies are desirable or efficient since these policies impose large losses on landowners and housing developers. The magnitude of these losses are hard to assess.9 However, our findings provide a compelling explanation of why affordable housing policies are increasingly popular among urban renters and populist politicians.

Finally, we study the impact of policies that progressively tax households that live in public housing, but have incomes that exceed the eligibility threshold. Note that approximately of all household living in public housing in Manhattan fall into that group. We show that we can design a revenue-neutral progressive tax system that can increase overall welfare by taxing ineligible high-income households at a substantially higher rate than eligible, low- and moderate-income households. Moreover, we can also use progressive taxation to reduce the average wait time for public housing and reallocate public housing to more deserving low-income households.

The rest of the paper is organized as follows. Section 2 discusses affordable housing policies in NYC and our data. Section 3 provides a new dynamic model of affordable housing markets. Section 4 discusses identification and estimation of the parameters of our model. Section 5 presents our empirical findings. Section 6 reports the findings from our welfare analysis and considers alternative housing policies. Section 7 offers our conclusions.

2 Data

Our empirical analysis focuses on the rental housing markets of NYC. Housing markets have been heavily regulated in NYC since the 1930s. As of 2011, over one million units were rent-stabilized representing roughly of the rental housing stock in NYC.10 Rent-stabilization generally applies to buildings of six or more units built between February 1, 1947, and December 31, 1973, and to those units that have exited from the rent-control program. Approximately of the city's stabilized units and nearly all stabilized units in buildings constructed after 1974 were voluntarily subjected to rent-stabilization by their owners in exchange for tax incentives from the city. Under the 421-a program, developers currently have to set aside of new apartments for poor and working-class tenants to receive tax abatements lasting 35 years.11

Involuntarily stabilized units, representing of the stabilized stock, are regulated based on a “housing emergency” declared by the city in 1974 and renewed every 3 years since. Under New York State's Rent-stabilization Law, the city may declare a housing emergency whenever the city's rental vacancy rate drops below . This law was most recently renewed in June 2015 and affects units with a maximum rent of . Rent stabilization sets maximum rates for annual rent increases. It also entitles tenants to have their leases renewed. The rent guidelines board meets every year to determine how much the landlord can set future rents on the lease.

In addition, low- and moderate-income households may have access to public housing. Providing adequate housing and shelter for low- and moderate-income households has been a policy goal of most federal, state, and city administrations in the United States since the passage of the Public Housing Act of 1937. The New York City Housing Authority (NYCHA) provides public housing and administers Section 8 housing vouchers for low- and moderate-income residents throughout the five boroughs of New York City. Households whose incomes do not exceed () of median income are eligible for the public housing program (voucher program). In addition, income limits are functions of family size. For example, in 2011 the income limit for a single person household was () while it was () for a family of four.

Applications for public housing are assigned a priority code based upon information that includes employment status, income, family size, and quality of the previous residence provided. Households are then placed on the housing authority's preliminary waiting list for an eligibility interview. Households are required to update or renew their applications every 2 years if they have not been scheduled for an interview. Upon passing the interview and background checks, applicants are then placed on a (borough-wide) waiting list.

More than New Yorkers reside in NYCHA's public housing apartments across the city's five boroughs. Another residents receive subsidized rental assistance in private homes through the NYCHA-administered Section 8 program. The NYCHA reported that families were on the waitlist for conventional public housing and families on the waitlist for Section 8. Little is known about the annual flows of waitlisted individuals into public housing. The NYT reported on July 23, 2013, that “the queue moves slowly. The apartments are so coveted that few leave them. Only 5400 to 5800 open up annually.” As of December 10, 2009, NYCHA stopped processing any new Section 8 applications due to the long waitlist. As a consequence, there is almost no mobility in and out of Section 8 housing markets. We, therefore, treat Section 8 housing as a completely separate market and focus on public housing and rent-stabilized housing in this paper.

The empirical analysis is based on the New York City Housing Vacancy Survey (NYCHVS) in 2011. The main advantage of this data set is that it matches households with units. It contains detailed information about both household characteristics and housing characteristics.

We focus on affordable housing for low- and moderate-income households which imposes three sample restrictions. First, we drop households whose average incomes exceed of median income level. This sample restriction is motivated by the fact that many high-income households are likely to own a condominium or house and, therefore, face a different choice set than low- and moderate-income households.12 Second, we drop all low-income households that receive vouchers since that market has been closed for at least 6 years.13 Finally, we drop all households not living in Manhattan since waitlists are operated at the borough level rather than city-wide. These restrictions reduce our sample size to 1666.14

Table 1 provides some descriptive statistics of the Manhattan housing market for 2011. Table 1 shows that a large fraction of the rental units in Manhattan is subject to rent-stabilization. The fraction was in 2011. At the same time, the average rent was in the unregulated market and in the regulated market.

Households tend to stay for long periods in their apartments. On average in 2011, households had occupied their apartments 16.18 years for public housing and 9.49 years for rent-stabilized housing. The turnover is much higher in the unregulated housing market. Not surprising, households in public housing are much poorer than households in rent-stabilized and unregulated housing. Households that live in public housing tend to be single-parent households, the majority headed by a female. Public housing households have more children, on average, than households in rent-stabilized or unregulated housing.

Finally, we rely on the 2008 Survey of Income and Program Participation (SIPP) to estimate the autocorrelation of the income process. Note that the NYHVS is a repeated cross-sectional data set and not a panel. Our SIPP sample consists of households that live in a metropolitan area in the state of NY.

3 A dynamic model of affordable housing markets

3.1 The baseline model

We consider a local housing market with three housing options: public housing (p), rent-regulated housing (r), and housing provided by the unregulated market (m). The exogenous housing stocks in public and rent-regulated housing are given by and .15

Time is discrete, . Households are infinitely lived and forward-looking. Households have a common discount factor β and maximize expected lifetime utility. In the baseline model, households only differ by income, denoted by y. Income evolves according to a stochastic law of motion that can be described by a stationary Markov process with transition density . Below we extend our model to allow for additional sources of household heterogeneity.

Changes in housing consumption are subject to adjustment costs. Households that either move across three housing sectors or change the level of housing within the unregulated market incur a utility cost of γ. The previous choice of the housing sector and the level of housing service in the unregulated market are, therefore, relevant state variables in this model.

The flow utility is defined over housing quality, h, and a numeraire good, b. Consider a household that rents in the unregulated market. Housing services can be purchased at price .16 Flow utility is, therefore, given by

(1)

Note that we are imposing the realistic assumption that low- and moderate-income households do not save and cannot borrow against uncertain future income. They are liquidity constrained and spend their incomes on housing and consumption goods in each period.

Households in the unregulated market can choose to stay in the same housing unit and continue to consume the housing service as in the previous period. The flow utility is given by

(2)

There are R discrete different levels of housing quality in the stabilized market. The flow utility associated with a rent-regulated unit of quality and price is given by

(3)

The next assumption captures the search frictions in that market.

Assumption 1.

• (a) Each period, there is a positive probability that a household receives an offer to move into a rent-regulated unit of quality .
• (b) Each household receives, at most, one offer per period.

The probabilities of receiving an offer to move into a stabilized housing unit are endogenous and depend on the supply and the voluntary outflow from regulated housing as discussed below in detail.

To simplify the notation, we set for the remainder of this section. All results can be easily generalized to account for heterogeneity in the quality and supply of rent-stabilized units.17 In our quantitative analysis below, we estimate a model with two discrete types of stabilized units.

Public housing provides a constant level of housing consumption, . Each household pays rent that is proportional to income. Hence, the rent is equal to τ y, where τ is the “income tax rate” associated with public housing. In practice, τ is typically 0.3 for most housing communities and cities. Per period utility in public housing is, therefore, given by

(4)

The local housing authority that administers the public housing program manages a waitlist. The priority score of a household is a monotonic function of the time spent on the waitlist. More formally, let w denote the time that a household has been on the waitlist. Let denote the probability that a household who has been on the waitlist for w periods receives an offer to move into public housing. The next assumption captures the behavior of the housing authority.

Assumption 2.

• (a) The housing authority makes take it or leave it offers. If a household rejects an offer, the priority score is reset to zero.
• (b) The outflow of public housing is voluntary, and the housing authority does not evict households from public housing.
• (c) Eligibility is determined by an income cut-off, denoted by and is checked every time period. Loss of eligibility means that the priority score is reset to zero.

These assumptions reflect a common practice of housing authorities in NYC and other U.S. metropolitan areas. Note that the distribution of priority scores is endogenous and determined in equilibrium as we discuss below.

The timing of decisions is as follows:

• 1. Each household gets a realization of income which determines the income distributions at the beginning of the period.
• 2. Some households get an offer to move into public housing. These offers are generated with probability .
• 3. Some households get an offer to move into rent-regulated housing unit of quality r. These offers are generated with probability .
• 4. Households decide whether to move or not. Households in the private sector determine their housing consumption level. All households obtain the flow utility that depends on their decisions.
• 5. Priority scores are updated.

Note that the flow utility is realized after households have relocated.

The four state variables in this model are the priority score, w, the income, y, the previous housing sector, d, and the previous level of housing service, . Define the conditional value functions associated with the four choices:

(5)

Note that we have imposed the condition that . Also note that is a deterministic function of the current income, y. We can derive recursive expressions for the unconditional value functions. The value functions, which are characterized in detail in Appendix A, then determine the optimal decision rules for each household.

To illustrate the optimal decision rules, we consider an estimated specification of the model with only one type of rent-stabilized housing. Figure 1 plots the policy function for a household in public housing. The vertical line indicates the income eligibility threshold for public housing. Optimal decision rules can be characterized by threshold rules. The dashed line indicates the decision rule of a household that received an offer to move into regulated housing while the dot-dashed line is the decision rule of a household without an offer.

Low- and moderate-income households prefer to live in public housing, moderate-income households prefer rent-regulated housing while high-income households prefer renting in the stabilized market, but prefer unregulated over public housing in the absence of stablized housing. Note that households in public housing are more likely to voluntarily move out of public housing if they have an offer to move into a rent-regulated apartment. Of course, the thresholds depend on the parameter values of the model. It is possible to generate specifications of the model in which high-income households want to live in unregulated housing. Similarly, we can characterize the decision rules for households in unregulated or rent-stabilized housing.

To finish characterizing the equilibrium, we need to define the inflows and outflows that are generated by our model. Let () denote the marginal distribution of wait times for households in unregulated (rent-regulated) housing in stationary equilibrium. Let denote the density of income of households that are inside public housing at the beginning of each period (before households have moved). Let denote the stationary density of current income and previous housing service conditional on wait time for households in the unregulated market. Similarly, let denote the stationary density of income conditional on wait time for households in the regulated market.18

Given these densities, we can characterize the resulting inflows and outflows for public and rent-stabilized housing. Consider the voluntary flow of households out of public housing. It is given by

(6)

Note that the first two terms are outflows to the unregulated market and the third term captures the outflow to the rent-regulated market. The flow into public housing is given by

(7)

The inflow from the unregulated market conditional on wait time is

(8)

The flows for rent-regulated housing, denoted by , which is given by

(9)

Similarly, we can characterize the outflow, denoted by , as discussed in detail in Appendix C of this paper.

In a stationary equilibrium, the inflows have to be equal to the outflows for public and rent-regulated housing.19

Definition 1.A stationary equilibrium for this model consists of the following: (a) offer probabilities and , (b) distributions , , , , and , and (c) value functions , , and , such that:

• 1. Households behave optimally and value functions satisfy the equations above.
• 2. The housing authority behaves according to the administrative rules described above.
• 3. The densities are consistent with the laws of motion and optimal household behavior.
• 4. satisfies the market clearing condition for public housing:
  (10)
• 5. satisfies the market clearing condition for rent-regulated housing:
  (11)

Finally, note that we can endogenize the price of housing in the unregulated market by assuming that that there is an upward sloping housing supply function . The demand for unregulated housing given by

(12)

Imposing the market clearing condition for unregulated housing then endogenizes . We use this approach in some of our counterfactual policy experiments.

Figure 2 illustrates the stationary equilibrium densities of income for a specification of our estimated model with one household type and only one type of rent-stabilized housing.

The top panel compares the marginal income densities of households in the unregulated market with densities of households that are in public housing.20 We find that households with a priority score of zero, who are ineligible for public housing, have a much higher income than those who live in public housing. Note that households with a priority score of 5 years have lower incomes, on average. This due to the fact that households with high priority scores must have had incomes below the eligibility threshold for a number of consecutive periods to remain eligible, while this criterion does not apply for households in public housing. The lower panel compares the income density of households in the rent-stabilized market with the income density of households in public housing. Again we find similar qualitative patterns. Households that live in stabilized housing with high priority scores look similar to households in public housing.

Next, we characterize the properties of equilibria with rationing. The main analytical result is summarized by the following proposition.

Proposition 1.Any stationary equilibrium with sufficiently strong excess demand for public housing has the property that there exists a value such that: , , and for all .

Note that implies that there are no households with priority score greater than , that is, , for .

Proof.We use a proof by contradiction. Suppose not, then

(13)

and next period there exists some households with priority score , hence which violates the stationarity definition and the definition of .

Suppose that and . This case violates the assumption that offers to households with lower priority ranks can only be made if all households with higher ranks receive offers.

Suppose that , and , then there will be no household in the next period which priority score which violates the stationarity assumption and that and that . □

The equilibrium has the property that everybody in the highest priority group obtains an offer to move into public housing. In addition, a fraction of the households with the second highest priority also gets an offer. The remaining households with the second highest priority score who do not get an offer this period will obtain an offer in the next period. The intuition for this result is the following. The waitlist partitions the potential demand into cohorts. By adjusting , we can smooth out the fraction of individuals that obtains an offer. Note that is not uniquely defined for . Since the housing authority makers take-it-or-leave-it offers, there will be no households with wait times larger than . Without loss of generality, we can, therefore, set for all .

Given this equilibrium offer function, the inflow into public housing has two components and is equal to

(14)

To finish the characterization of the equilibrium, we need to provide the laws of motion for the equilibrium densities. Equations (21)–(29) in Appendix B provide the details.

4 Identification and estimation

Since equilibria can only be computed numerically, we introduce a parametrization of the model in this section. We then discuss identification and estimation.

There are four choices in our model: public housing p, rent-stabilized housing r, move to a new unregulated unit m, stay in the same unregulated unit s. We assume that the flow utility functions can be approximated by a Stone–Geary utility function. Hence, the four flow utilities are given by:

(15)

Recall that α determines the housing share parameter if . The parameter captures minimum housing consumption. Adding this parameter, therefore, implies that our model predicts that housing shares are decreasing in income. Both parameters are, therefore, identified based on the observed housing shares by income.

We set the annual discount factor, β, at 0.95. We normalize the price of housing in the unregulated market, , to be equal to one since the units of housing services are arbitrary. The tax rate in public housing, τ, is determined by the administration of public housing programs. It is a state policy that renters in public housing pay of their income in rent.23

To identify and estimate the price discount in the rent-regulated market, , we assume that market rents can be decomposed into a price and a quality index. We assume that the quality index is the same for units in the unregulated and the regulated markets, but the prices are not. We can, therefore use the techniques discussed in Sieg, Smith, Banzhaf, and Walsh (2002) to identify and estimate the price discount in the regulated market.

The quality parameters for are identified based on our classification algorithm discussed below and the observed market rents for each type of unit type conditional on observed characteristics.

Public housing quality, , is identified from the observed demand for public housing, such as average income of households in the public housing and average time spent in public housing. As can be seen in Figure 1, a household whose income is below a certain cutoff chooses public housing if offered. The cutoff is an increasing function of the public housing quality. The model predicts that the average income of households in public housing is an increasing function of the public housing quality. Similarly, with the higher cutoff, households are less likely to move out of the public housing, thus average time spent in the unit is also an increasing function of the public housing quality.

Income consists of a fixed minimum income plus a stochastic component, . Hence, we have . We assume that the logarithm of stochastic income for each household follows an process:

(16)

where the standard deviation of the error term is given by σ. The mean and the standard deviation of log income are identified of the observed income distributions in the data. We use panel data from the Survey of Income and Program Participation to estimate the autocorrelation parameter of the income process.

All parameters of the income process and household preferences depend on household type in the extended model. The household type is observed by the econometrician. The identification argument, therefore, extends to that model since all relevant moments are observed conditional on type.

The introduction of moving costs generates a more realistic model. Previous empirical housing studies have suggested that moving costs may be substantial. Adding these frictions generates persistence in housing choices in the unregulated market. As a consequence, the moving cost parameter, γ, is primarily identified from the observed length of stay in an apartment that is not subject to rent-stabilization.

The arguments for identification are constructive and suggest that we can estimate the parameters of our model using a method of moments estimator. We use the following moments in estimation: the fraction of each housing type, the average time spent in the unit by housing type, the average income by housing type, the variance of income by housing type, the average rents by housing type. Asymptotic standard errors can be consistently estimated using the standard formula for a parametric method of moments estimator provided, for example, in Newey and McFadden (1994).

5 Empirical results

5.2 Parameter estimates and goodness of fit

Table 3 reports the estimated parameters and standard errors for three specifications of our model. First, we estimate the baseline model. Second, we add heterogeneity in regulated housing types. Finally, we add heterogeneity in discrete household types. Overall, we find that all parameters are estimated with good precision.

Table 3. Estimated parameters.

	I	II	III
	Baseline	1 Household type	2 Household type
	all	all	female	male
α	0.23 (0.007)	0.28 (0.009)	0.25 (0.016)	0.30 (0.019)
	12,395 (1270)	10,480 (149)	10,933 (43)	10,439 (45)
γ	14,014 (40)	11,081 (390)	10,759 (1402)	14,078 (1883)
μy	9.62 (0.038)	9.74 (0.020)	9.80 (0.031)	9.79 (0.033)
σ	1.12 (0.007)	1.07 (0.008)	1.06 (0.015)	1.01 (0.016)
ρ	0.67 (0.02)	0.67 (0.02)	0.63 (0.03)	0.70 (0.03)
hp	17,912 (169)	18,069 (203)	(702)
hr	32,172 (219)
		27,499 (378)	(475)
		34,467 (290)	(513)

• Note: Standard errors are in parentheses.

Consider the baseline model in Column I. The parameter α captures the housing expenditure share for households with sufficiently high income such that the impact of the minimum housing consumption is negligible. In our application, these are household with incomes of approximately and more. Low-income households tend to have higher housing expenditure shares than high-income households. This is captured by our estimate of , which is approximately . Once we allow for heterogeneity in regulated housing types in Column II, our estimate of α increases to 0.28, and the estimate decreases to . As a consequence, low- and moderate-income households spend approximately 40 to of income on housing if they rent in the unregulated market. High-income households spend approximately 30– of income on housing. We also find that there are substantial moving costs, which are captured by the parameter γ. The estimate is () in Column I (II).

Column III shows that female-headed households have slightly lower estimate of α and higher estimate of than male-headed households. Female-headed households also tend to have lower moving costs than male-headed households. None of these differences are, however, significant at conventional levels. The parameters of the income process, however, depend on the observed household type. Female-headed households have more volatile and less persistent incomes than male-headed households.

Housing quality is measured as equivalent expenditures in the unregulated market. The baseline estimate in Column I shows that an average public housing unit in Manhattan provides the same quality as a unit that rents for in the unregulated market. The average quality of rent-stabilized housing in the baseline model is . Allowing for heterogeneity in rent-stabilized housing in Column II indicates that the quality for a low (high) quality rent-stabilized apartment is (). Allowing for heterogeneity in household types does not affect these estimates. We thus conclude that public housing units are of significantly lower quality than low-quality, rent-stabilized housing units. High-quality, rent-stabilized units provide, on average, more housing services than low-quality units.

Table 4 summarizes some properties that correspond to the equilibria that are implied by the parameter estimates. Our model generates wait times of approximately 19 to 20 years. The probability of finding a rent-stabilized unit is approximately for high-quality units and for low-quality units. Overall, these predictions are quite plausible.

Table 4. Properties of equilibrium.

		Baseline	1 Household type	2 Household type
wait		20	19	18
times		0.77	0.19	0.01
search	q1	0.26	0.11	0.10
frictions	q2		0.26	0.23

Our estimates imply that affordable housing is an attractive option for low- and moderate-income households in Manhattan. To illustrate the magnitude of these effects, we compute compensating variations that measure the benefits of having access to rent-stabilized housing relative to renting in the unregulated market. Figure 3 plots the compensating variations by income for the two quality levels of rent-stabilized housing.

Our analysis suggests that even low-quality units of affordable housing are quite attractive. A household with approximately gains approximately from having access to a low-quality, rent-stabilized units. High-quality units are also attractive. A household with approximately gains approximately from having access to a high-quality, rent-stabilized unit. Figure 3 shows that there is an inverted-U shaped relationship between income and welfare gains. As a consequence, rent-stabilization policies create a fair amount of misallocations in housing markets.25

Given these large benefits associated with affordable housing, it is not surprising that rent-stabilization and affordable housing policies are popular at the ballot box, not only with low- and moderate-income households, but also with higher-income households that rent. As a consequence, our model explains the prevalence and political popularity of affordable housing policies in places such as NYC and Los Angeles.

We focused in this section on the Manhattan subsample. We also estimated specifications of the model using the full sample that includes renters of all five boroughs. The results are qualitatively similar to the results reported above. The main difference is that the price discount for affordable housing, wait and search times are lower using data from all five boroughs.

Table 5 reports a variety of goodness of fit statistics. We report the key statistics for the models in Columns I and II. Overall, we find that our model fits the key moments used in estimation well.26

Table 5. Model fit.

	Years		Income		Market rent
Housing	data	model	data	model	data	model	Income rage
Baseline
Public	16.18	17.57	32,930	30,405	—	—
Regulated	9.49	9.41	54,739	57,983	1317	1307
Market	3.85	2.64	71,045	70,045	—	—
					2581	2444	75,000 ≤ y < 100,000
					2895	2951	100,000 ≤ y < 125,000
					3430	3477	125,000 ≤ y < 150,000
1 Household type
Public	16.18	16.81	32,930	28,872	—	—
Regulated1	9.18	10.04	49,651	49,326	1071	1117
Regulated2	9.81	6.13	59,859	63,588	1570	1400
Market	3.85	2.17	71,045	69,673	—	—
					2581	2627	75,000 ≤ y < 100,000
					2895	3088	100,000 ≤ y < 125,000
					3430	3655	125,000 ≤ y < 150,000

6 Policy analysis

6.1 Increasing the supply of rent-stabilized housing

The popularity of affordable policies is in stark contrast to longer term trends in the supply of affordable housing in NYC. Landlords primarily bear the burden of the rent-stabilization policies. Note that less than of all apartments in NYC are voluntarily rent-stabilized. These landlords obtain significant tax breaks as a return for making a fraction of the housing units affordable. The vast majority of rental units in NYC—approximately —are “involuntarily” stabilized under New York state's rent-stabilization law. In NYC, landlords have long been allowed to deregulate vacant apartments if the legal rent for a new renter exceeds a threshold, which is currently a month. Between 1993 and 2015, more than apartments have been converted to market rates through vacancy decontrol which has led to a significant decline in the supply of affordable housing (WSJ, 2015).27

Not surprisingly, these trends have not gone unnoticed. As a candidate, the current mayor of NYC, Bill de Blasio, successfully ran on a platform that promised significant increases in the provision of affordable housing. Once in office, he proposed and the city council recently adopted a 10-year plan to build and retain affordable housing units in the NYC area through various rezoning laws. We can use our model to simulate the effects of these types of policy changes on renters' welfare. Using our model with two affordable housing types, we increase the supply of affordable housing by up to . We consider the impact of this policy change on the probability of finding an affordable housing unit, the wait time for public housing, as well as the distribution of renters' welfare in the economy.

Here, we consider the impact of increasing the supply of regulated housing from to of total rental units in Manhattan. We increase the supply of low-quality and high- quality regulated housing equally. Figure 4 shows that a increase in the supply of affordable housing substantially increases the probability of receiving an offer to move into a regulated housing unit in equilibrium. The probability of finding a low-quality unit increases from 10 to , while the probability of finding a high-quality unit increases from 25 to . Wait times for public housing also decrease by up to 1 year.

The reduced wait and search times are associated with a more efficient allocation of public and rent-stabilized housing in equilibrium. Households are more likely to move out of affordable units when they receive positive income shocks. Hence, those units can be reallocated faster to more needy households. As a consequence, the time spent in public or regulated housing decreases significantly. Similarly, households in the unregulated market also spend less time in the unit because of the reduced wait and search times for affordable housing.28

In a static framework, subsidized housing is primarily occupied by low-income households. An increase in the supply of subsidized housing implies that the average income of households in those subsidized units increases as more moderate-income households move into those units. In our dynamic framework, there is another countervailing effect. As the allocation of affordable housing becomes more efficient, those units are occupied by households whose realized income is low due to negative income shocks. As we increase the supply of regulated housing, we find that the average income of households in public housing and low-quality regulated housing decreases.

We can measure the welfare gains for households using compensating variations. Figure 5 plots the average welfare gain as a function of the increase in the regulated housing stock. We find that the average welfare gain of a increase in affordable housing is approximately . Note that all households in our model benefit from a permanent increase in rent-stabilized housing. The lower wait and search times imply that all households are better insulated against negative future income shocks. Overall, the welfare gains from this insurance are larger for high-income households. Of course, the biggest gains are for those who prefer rent-stabilized housing and now are more likely to obtain access to it.

Finally, we evaluate how sensitive the results of our policy analysis are to changes in the supply of unregulated housing. In the baseline model, we assume that the housing supply for unregulated housing is perfectly elastic. Hence, prices in unregulated housing are not affected by the change in the supply of affordable housing. Alternatively, we can use an aggregate housing supply function given by . We set the constant l such that demand and supply are equal when in our baseline year 2011. To evaluate the robustness of our findings, we repeated the exercise above assuming a variety of different values for the supply elasticity of unregulated housing. Here, we focus on the case when . A increase in regulated housing reduces the demand for unregulated housing. As a consequence, the rental price for unregulated housing drops by . As private housing gets cheaper, it becomes more attractive. As a consequence, the reduction in waiting and search times are even steeper than in the baseline model. But overall, we obtain the same qualitative and quantitative results.29

6.2 Taxing ineligible households in public housing

One notable feature of existing public housing policies is that housing authorities rarely ask a household to leave public housing once its income exceeds the eligibility threshold. Approximately of households living in public housing in Manhattan have incomes that exceed of the median income and are, thus, likely to be ineligible of housing aid. For these households, housing aid is de facto an open-ended entitlement program.

Since evictions are politically not feasible, we consider a different mechanism to encourage higher income households to leave public housing. Currently, the housing authority uses a flat income tax to generate rental income. The tax rate is . The basic idea is to replace the current flat tax with a revenue-neutral progressive tax system. Here, we explore a piecewise linear tariff with a low marginal rate, denoted by τ, for incomes below the eligibility threshold () and a higher rate, denoted by , for incomes above the threshold. Hence, the new tax function is given by

(17)

For a given value of τ, we can solve for the revenue-neutral value of , using a simple line-search algorithm.

Figure 6 illustrates revenue-neutral combinations of and τ in equilibrium. Figure 7 captures the welfare implications of these policy changes. Overall, we find that the progressive revenue-neutral tax is welfare improving. Welfare gains are up to per household when we set and . The more progressive the tax system, the higher the aggregate welfare gains.

However, these revenue-neutral policies do not reduce wait times. The main reason for this result is that the progressive tax schedule provides even stronger insurance against negative income shocks. Thus, anticipating future negative income shocks, households are even less willing to move out of public housing.

To reduce wait times, we need to consider policies that are not revenue-neutral. For example, we can fix τ at the current level of 0.3 and then increase . This policy makes public housing less attractive for low- and moderate-income households. It reduces the average waiting times and generates additional revenues. However, it does not necessarily improve welfare. For example, if we set , we can reduce the waiting time of the public housing by 1.1 years. However, the average welfare of the renters will drop by on average.

7 Conclusions

We have developed a new dynamic model that captures search and queuing frictions in the rental markets for affordable housing, as well as mobility cost. We have characterized the stationary equilibrium with rationing that arises in the model.30 We have shown how to identify and estimate the structural parameters of the model. Our application focuses on the housing markets of Manhattan in 2011. Overall, our model fits the observed sorting of households well. We have characterized the distribution of welfare that arises in our model. We have shown that access to low (high) quality of affordable housing can increase welfare by as much as (). As a consequence, our model provides a compelling explanation of why affordable housing policies have been popular with the vast majority of urban renters in NYC. Finally, we have studied the effects of expanding the supply of affordable housing and the effects of progressively taxing households in public housing. We find that both policies can improve overall welfare.

We should point out that we cannot conclude from this analysis that affordable housing policies such as those in NYC are desirable. First, our analysis does not allow us to measure the costs that are imposed on landlords. Clearly, these policies primarily redistribute wealth and income from landlords to renters. The magnitude of the welfare losses imposed on landlords is largely unknown. Second, rent stabilization policies weaken the incentive to invest in housing. As a consequence, these policies have a significant negative impact on the long-term supply of both regulated and unregulated housing.

The main focus of this paper is on positive analysis. We provide new estimates of the benefits that a renter obtains from living in an affordable or public housing unit. Our analysis provides a compelling explanation of why affordable housing policies are popular with the vast majority of urban renters and populist politicians. From the perspective of landowners and housing developers, affordable housing policies are undoubtedly costly. The main political advantage of affordable housing policies is that they allow local politicians to finance redistribution by implicitly taxing landlords and housing developers that benefit from rezoning policies. Since housing developers and landowners tend to benefit the most from improvements in urban quality via capitalization effects, affordable housing policies effectively redistribute part of these gains of urban redevelopment and improvement to low- and moderate-income renters. These renters tend to be the majority of voters in city elections.